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Current Question (ID: 19107)

Question:

$\text{A particle is projected at an angle of } 30^\circ \text{ with the ground at a speed of } 40 \text{ m/s. The speed of the particle after two seconds is: (use } g = 10 \text{ m/s}^2)$

Options:

1. $20\sqrt{2} \text{ m/s}$
2. $20\sqrt{3} \text{ m/s}$
3. $20 \text{ m/s}$
4. $10\sqrt{3} \text{ m/s}$

Solution:

$\text{Hint: At } t = 2 \text{ s, particle will be at maximum height.}$ $\text{Step: Find the speed of the particle after two seconds.}$ $\text{A particle is projected with an initial velocity } 40 \text{ m/s making an angle } 30^\circ.$ $\text{The speed of the particle is given by: } \vec{v} = \vec{u} + \vec{a}t$ $\Rightarrow \vec{v} = (u_x \hat{i} + u_y \hat{j}) - gt \hat{j}$ $u_x = 40 \cos 30^\circ = 20\sqrt{3} \text{ m/s}$ $u_y = 40 \sin 30^\circ = 20 \text{ m/s}$ $\Rightarrow \vec{v} = (20\sqrt{3} \hat{i} + 20 \hat{j}) - (10 \times 2) \hat{j} \text{ m/s}$ $\Rightarrow \vec{v} = 20\sqrt{3} \hat{i} \text{ m/s}$ $\text{So, after } t = 2 \text{ s, the particle will be at maximum height having velocity } 40 \cos 30^\circ \text{ m/s.}$ $\text{Therefore, the speed of the particle after two seconds is } 20\sqrt{3} \text{ m/s.}$ $\text{Hence, option (2) is the correct answer.}$

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}